Olympiad problems

2011 JBMO Shortlist, 4

Tags: geometry , JBMO

Point ${D}$ lies on the side ${BC}$ of $\vartriangle ABC$. The circumcenters of $\vartriangle ADC$ and $\vartriangle BAD$ are ${O_1}$ and ${O_2}$, respectively and ${O_1O_2\parallel AB}$. The orthocenter of $\vartriangle ADC$is ${H}$ and ${AH=O_1O_2}.$ Find the angles of $\vartriangle ABC$ if $2m\left( \angle C \right)=3m\left( \angle B \right).$

2008 JBMO Shortlist, 7

Tags: JBMO , geometry

Let $ABC$ be an isosceles triangle with $AC = BC$. The point $D$ lies on the side $AB$ such that the semicircle with diameter $BD$ and center $O$ is tangent to the side $AC$ in the point $P$ and intersects the side $BC$ at the point $Q$. The radius $OP$ intersects the chord $DQ$ at the point $E$ such that $5 \cdot PE = 3 \cdot DE$. Find the ratio $\frac{AB}{BC}$ .

2023 JBMO Shortlist, N1

Tags: number theory , JBMO , JBMO Shortlist

Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$. [i]Nikola Velov, North Macedonia[/i]

2011 JBMO Shortlist, 9

Tags: JBMO , algebra

Let $x_1,x_2, ..., x_n$ be real numbers satisfying $\sum_{k=1}^{n-1} min(x_k; x_{k+1}) = min(x_1; x_n)$. Prove that $\sum_{k=2}^{n-1} x_k \ge 0$.

2021 Junior Balkаn Mathematical Olympiad, 3

Tags: geometry , JBMO , JBMO Shortlist

Let $ABC$ be an acute scalene triangle with circumcenter $O$. Let $D$ be the foot of the altitude from $A$ to the side $BC$. The lines $BC$ and $AO$ intersect at $E$. Let $s$ be the line through $E$ perpendicular to $AO$. The line $s$ intersects $AB$ and $AC$ at $K$ and $L$, respectively. Denote by $\omega$ the circumcircle of triangle $AKL$. Line $AD$ intersects $\omega$ again at $X$. Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.

2015 JBMO Shortlist, A5

Tags: inequalities , JBMO , China , BPSQ , JBMO Shortlist , High school olympiad

The positive real $x, y, z$ are such that $x^2+y^2+z^2 = 3$. Prove that$$\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2$$

2008 JBMO Shortlist, 3

Tags: JBMO , system of equations , algebra

Let the real parameter $p$ be such that the system $\begin{cases} p(x^2 - y^2) = (p^2- 1)xy \\ |x - 1|+ |y| = 1 \end{cases}$ has at least three different real solutions. Find $p$ and solve the system for that $p$.

2020 JBMO Shortlist, 1

Tags: algebra , system of equations , Junior Balkan , polynomial , Junior , JBMO , jbmo2020

Find all triples $(a,b,c)$ of real numbers such that the following system holds: $$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

2023 JBMO Shortlist, N5

Tags: JBMO , JBMO Shortlist , number theory

Find the largest positive integer $k$ such that we can find a set $A \subseteq \{1,2, \ldots, 100 \}$ with $k$ elements such that, for any $a,b \in A$, $a$ divides $b$ if and only if $s(a)$ divides $s(b)$, where $s(k)$ denotes the sum of the digits of $k$.

2012 Junior Balkan MO, 1

Tags: JBMO , algebra , Inequality , JBMO Shortlist

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that \[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\] When does equality hold?

2019 JBMO Shortlist, A4

Tags: algebra , Junior , JBMO , 2019 , Balkan , inequalities

Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $a^4 - 2019a = b^4 - 2019b = c$. Prove that $- \sqrt{c} < ab < 0$.

2007 JBMO Shortlist, 4

Tags: JBMO , geometry

Let $S$ be a point inside $\angle pOq$, and let $k$ be a circle which contains $S$ and touches the legs $Op$ and $Oq$ in points $P$ and $Q$ respectively. Straight line $s$ parallel to $Op$ from $S$ intersects $Oq$ in a point $R$. Let $T$ be the intersection point of the ray $PS$ and circumscribed circle of $\vartriangle SQR$ and $T \ne S$. Prove that $OT // SQ$ and $OT$ is a tangent of the circumscribed circle of $\vartriangle SQR$.

2023 Junior Balkan Mathematical Olympiad, 1

Tags: number theory , JBMO , JBMO Shortlist

Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$. [i]Nikola Velov, North Macedonia[/i]

2024 Junior Balkan MO, 2

Tags: geometry , circumcircle , collinear , Balkan , JBMO

Let $ABC$ be a triangle such that $AB < AC$. Let the excircle opposite to A be tangent to the lines $AB, AC$, and $BC$ at points $D, E$, and $F$, respectively, and let $J$ be its centre. Let $P$ be a point on the side $BC$. The circumcircles of the triangles $BDP$ and $CEP$ intersect for the second time at $Q$. Let $R$ be the foot of the perpendicular from $A$ to the line $FJ$. Prove that the points $P, Q$, and $R$ are collinear. (The [i]excircle[/i] of a triangle $ABC$ opposite to $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.) [i]Proposed by Bozhidar Dimitrov, Bulgaria[/i]

2008 JBMO Shortlist, 8

Tags: JBMO , inequalities , algebra

Show that $(x + y + z) \big(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\big) \ge 4 \big(\frac{x}{xy+1}+\frac{y}{yz+1}+\frac{z}{zx+1}\big)^2$ , for all real positive numbers $x, y $ and $z$.

2011 JBMO Shortlist, 4

Tags: JBMO , combinatorics , Construct

In a group of $n$ people, each one had a different ball. They performed a sequence of swaps, in each swap, two people swapped the ball they had at that moment. Each pair of people performed at least one swap. In the end each person had the ball he/she had at the start. Find the least possible number of swaps, if: a) $n = 5$, b) $n = 6$.

2016 JBMO Shortlist, 4

Tags: JBMO , combinatorics , graph theory , planar graph , triangulation

A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \ge 3$, determine the largest integer $m$ such that no planar $n$-gon splits into less than $m$ triangles.

2022 Junior Balkan Mathematical Olympiad, 1

Tags: algebra , JBMO , JBMO Shortlist

Find all pairs of positive integers $(a, b)$ such that $$11ab \le a^3 - b^3 \le 12ab.$$

2015 JBMO Shortlist, 3

Tags: geometry , JBMO

Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$ , say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$ , say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$. Evangelos Psychas (Greece)

2016 JBMO Shortlist, 2

Tags: geometry , JBMO

Let ${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$. Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the triangles ${ADE}$and ${BOC}$ are tangent to each other.

2019 Junior Balkan MO, 4

Tags: combinatorics , Junior , Balkan , JBMO , 2019

A $5 \times 100$ table is divided into $500$ unit square cells, where $n$ of them are coloured black and the rest are coloured white. Two unit square cells are called [i]adjacent[/i] if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of $n$.

2016 JBMO Shortlist, 2

Tags: inequalities , JBMO

Let $a,b,c $be positive real numbers.Prove that $\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$.

2011 JBMO Shortlist, 2

Tags: JBMO , combinatorics

Can we divide an equilateral triangle $\vartriangle ABC$ into $2011$ small triangles using $122$ straight lines? (there should be $2011$ triangles that are not themselves divided into smaller parts and there should be no polygons which are not triangles)

2008 JBMO Shortlist, 6

Tags: JBMO , algebra , inequalities

If the real numbers $a, b, c, d$ are such that $0 < a,b,c,d < 1$, show that $1 + ab + bc + cd + da + ac + bd > a + b + c + d$.

2003 JBMO Shortlist, 2

Tags: geometry , JBMO , area

Is there a triangle with $12 \, cm^2$ area and $12$ cm perimeter?